3D Builder won't accept meshes made from Particle System in Cycles Render
Looking at the reference image I'd say it could be better to model that hair via simple mesh without any particle system. This way is often used for hair in some styles of 3d and 2d art, it allows to avoid possible slowdowns of particle system as well.

As to converted particles probably you used Path option as how particles are rendered in the Render layout of the particle system settings. 



After applying Particle System modifier they become simple lines (edges with no thickness) which can't be printed because they are non-manifold geometry. 

One way is to make that geometry manifold is to solidify it using Skin modifier on converted particles.  

Select object with converted particles (note that you might want to leave particle system undeleted but hidden in case some changes are needed). 
Add Skin modifier to it. By default modifier will solidify only one of converted particles because it marked a vertex as root for only one edge while it needs one root vertex for all the connected pieces of geometry. As per manual:


  Mark Root
      Marking a vertex as root causes that vertex to be used for calculating     rotations for connected limbs. [...]
      Each set of connected vertices should have one root node.


The easiest way to mark one vertex for multiple edges of the converted particles is to enter Edit mode, select all and press Mark Root button. It will find one vertex itself to mark so no need to deselect anything.
Once done all the particles will have thickness and become manifold, all-quads geometry. Make sure to apply modifier if exporting mesh.




Another simple and working a bit differently solution is to 
set Object as how particles are rendered in Render rollout instead and give it some thin and long cube as object to be used as hair particle to make particles based on it. See underlined option on the 1-st screenshot, change it to Object and choose cube you created. No need to make follow it anything, it will be done automatically.  

Make sure to rotate base mesh accordingly so it's oriented accordingly (e.g. outside of the mesh, typically you will need to rotate it along X axis).  

This way is less flexible and gives slightly different result as it doesn't allow e.g. Brownian Motion setup to change particle orientation along its local X and Y axis, only along Z, and other options which allow to randomize  particles.